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A bstract We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in the light-ray operators that appear in the stress-tensor OPE. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in N = 4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).

A bstract We consider weakly-coupled QFT in AdS at finite temperature. We compute the holographic thermal two-point function of scalar operators in the boundary theory. We present analytic expressions for leading corrections due to local quartic interactions in the bulk, with an arbitrary number of derivatives and for any number of spacetime dimensions. The solutions are fixed by judiciously picking an ansatz and imposing consistency conditions. The conditions include analyticity properties, consistency with the operator product expansion, and the Kubo-Martin-Schwinger condition. For the case without any derivatives we show agreement with an explicit diagrammatic computation. The structure of the answer is suggestive of a thermal Mellin amplitude. Additionally, we derive a simple dispersion relation for thermal two-point functions which reconstructs the function from its discontinuity.

A bstract We study propagation of a probe particle through a series of closely situated gravitational shocks. We argue that in any UV-complete theory of gravity the result does not depend on the shock ordering — in other words, coincident gravitational shocks commute. Shock commutativity leads to nontrivial constraints on low-energy effective theories. In particular, it excludes non-minimal gravitational couplings unless extra degrees of freedom are judiciously added. In flat space, these constraints are encoded in the vanishing of a certain “superconvergence sum rule.” In AdS, shock commutativity becomes the statement that average null energy (ANEC) operators commute in the dual CFT. We prove commutativity of ANEC operators in any unitary CFT and establish sufficient conditions for commutativity of more general light-ray operators. Superconvergence sum rules on CFT data can be obtained by inserting complete sets of states between light-ray operators. In a planar 4d CFT, these sum rules express a − c c in terms of the OPE data of single-trace operators.

A bstract We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a “thermal inversion formula” whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical O ( N ) model at leading order in 1 /N . Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.

A bstract The extent to which quantum mechanical features of black holes can be understood from the Euclidean gravity path integral has recently received significant attention. In this paper, we examine this question for the calculation of the supersymmetric index. For concreteness, we focus on the case of charged black holes in asymptotically flat four-dimensional N = 2 ungauged supergravity. We show that the gravity path integral with supersymmetric boundary conditions has an infinite family of Kerr-Newman classical saddles with different angular velocities. We argue that fermionic zero-mode fluctuations are present around each of these solutions making their contribution vanish, except for a single saddle that is BPS and gives the expected value of the index. We then turn to non-perturbative corrections involving spacetime wormholes and show that a fermionic zero mode is present in all these geometries, making their contribution vanish once again. This mechanism works for both single- and multi-boundary path integrals. In particular, only disconnected geometries without wormholes contribute to the gravitational path integral which computes the index, and the factorization puzzle that plagues the black hole partition function is resolved for the supersymmetric index. Finally, we classify all other single-centered geometries that yield non-perturbative contributions to the gravitational index of each boundary.

A bstract We estimate thermal one-point functions in the 3d Ising CFT using the operator product expansion (OPE) and the Kubo-Martin-Schwinger (KMS) condition. Several operator dimensions and OPE coefficients of the theory are known from the numerical bootstrap for flat-space four-point functions. Taking this data as input, we use a thermal Lorentzian inversion formula to compute thermal one-point coefficients of the first few Regge trajectories in terms of a small number of unknown parameters. We approximately determine the unknown parameters by imposing the KMS condition on the two-point functions 〈 σσ 〉 and 〈 ϵϵ 〉. As a result, we estimate the one-point functions of the lowest-dimension ℤ 2 -even scalar ϵ and the stress energy tensor T μν . Our result for 〈 σσ 〉 at finite-temperature agrees with Monte-Carlo simulations within a few percent, inside the radius of convergence of the OPE.

A bstract We initiate a study of asymptotic detector operators in weakly-coupled field theories. These operators describe measurements that can be performed at future null infinity in a collider experiment. In a conformal theory they can be identified with light-ray operators, and thus have a direct relation to the spectrum of the theory. After a general discussion of the underlying physical picture, we show how infrared divergences of general detector operators can be renormalized in perturbation theory, and how they give rise to detector anomalous dimensions. We discuss in detail how this renormalization can be performed at the intersections of the Regge trajectories where non-trivial mixing occurs, which is related to the poles in anomalous dimensions at special values of spin. Finally, we discuss novel horizontal trajectories in scalar theories and show how they contribute to correlation functions. Our calculations are done in the example of ϕ 4 theory in d = 4 − ϵ dimensions, but the methods are applicable more broadly. At the Wilson-Fisher fixed point our results include an explicit expression for the Pomeron light-ray operator at two loops, as well as a prediction for the value of the Regge intercept at five loops.

A bstract We study a product of null-integrated local operators O 1 and O 2 on the same null plane in a CFT. Such null-integrated operators transform like primaries in a fictitious d − 2 dimensional CFT in the directions transverse to the null integrals. We give a complete description of the OPE in these transverse directions. The terms with low transverse spin are light-ray operators with spin J 1 + J 2 − 1. The terms with higher transverse spin are primary descendants of light-ray operators with higher spins J 1 + J 2 − 1 + n , constructed using special conformally-invariant differential operators that appear precisely in the kinematics of the light-ray OPE. As an example, the OPE between average null energy operators contains light-ray operators with spin 3 (as described by Hofman and Maldacena), but also novel terms with spin 5, 7, 9, etc. These new terms are important for describing energy two-point correlators in non-rotationally-symmetric states, and for computing multi-point energy correlators. We check our formulas in a non-rotationally-symmetric energy correlator in N = 4 SYM, finding perfect agreement.

A bstract We analyze the classical and quantum vacua of 2d N = 8 8 supersymmetric Yang-Mills theory with SU( N ) and U( N ) gauge group, describing the worldvolume interactions of N parallel D1-branes with flat transverse directions ℝ 8 . We claim that the IR limit of the SU( N ) theory in the superselection sector labeled M (mod N ) — identified with the internal dynamics of ( M , N )-string bound states of the Type IIB string theory — is described by the symmetric orbifold N = 8 8 sigma model into ℝ 8 D − 1 / S D when D = gcd( M , N ) > 1, and by a single massive vacuum when D = 1, generalizing the conjectures of E. Witten and others. The full worldvolume theory of the D1-branes is the U( N ) theory with an additional U(1) 2-form gauge field B coming from the string theory Kalb-Ramond field. This U( N ) + B theory has generalized field configurations, labeled by the ℤ -valued generalized electric flux and an independent ℤ N -valued ’t Hooft flux. We argue that in the quantum mechanical theory, the ( M , N )-string sector with M units of electric flux has a ℤ N -valued discrete θ angle specified by M (mod N ) dual to the ’t Hooft flux. Adding the brane center-of-mass degrees of freedom to the SU( N ) theory, we claim that the IR limit of the U( N ) + B theory in the sector with M bound F-strings is described by the N = 8 8 sigma model into S y m D ℝ 8 . We provide strong evidence for these claims by computing an N = 8 8 analog of the elliptic genus of the UV gauge theories and of their conjectured IR limit sigma models, and showing they agree. Agreement is established by noting that the elliptic genera are modular-invariant Abelian (multi-periodic and meromorphic) functions, which turns out to be very restrictive.

The majority of this thesis is dedicated to certain nonlocal aspects of conformal field theories (CFTs). Two main directions are the study of CFTs on a particular globallynontrivial spacetime, and the study of particular nonlocal observables in CFTs.The first aspect concerns with the study of CFTs on a spacetime with imaginary periodic time, equivalent to the study of static properties of a CFT at finite temperature. We introduce bootstrap techniques for determining finite-temperature data of CFTs, and make predictions for the 2+1-dimensional O(N) model at large N and the 2+1-dimensional Ising model.The second aspect is the study of light-ray operators in CFTs — operators that are supported on light-like trajectories. We propose the “stringy equivalence principle,” stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein’s General Relativity that must hold in all consistent theories of gravity. Analyzing properties of light-ray operators dual to gravitational shocks, we prove the stringy equivalence principle for holographic CFTs dual to gravity in Anti-de Sitter (AdS) spacetimes. We place stringent constraints on effective theories of gravity. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors — such as calorimeters. We use the light-ray OPE to compute event shape observables suitable for conformal collider physics in 3+1-dimensional N= 4 super-Yang-Mills Theory.An additional part of this thesis determines the low energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy — a modified thermal partition function named the refined elliptic genus — we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p, q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.